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The Lanczos algorithm, used in computational physics, faces numerical issues. Numerical Lanczos vectors escape true vector spaces, threatening interpretations of operator growth in Krylov complexity.

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Area of Science:

  • Computational physics
  • Numerical analysis
  • Quantum mechanics

Background:

  • The Lanczos algorithm is a long-standing method in computational physics, primarily used for approximating extreme eigenvalues and eigenvectors.
  • Recent interest has focused on the Lanczos algorithm's basis vectors within the context of Krylov complexity.
  • While numerically stable for eigenvalue approximation, the algorithm presents challenges for Krylov basis construction.

Purpose of the Study:

  • To investigate the numerical instabilities encountered when constructing the Krylov basis using the Lanczos algorithm.
  • To demonstrate that standard reorthogonalization methods are insufficient to address these numerical problems.
  • To explain the observed deviation of numerical Lanczos vectors from the exact vector space.

Main Methods:

  • Analysis of numerical precision effects on the Lanczos algorithm.
  • Theoretical investigation into the behavior of Lanczos vectors in finite-precision arithmetic.
  • Comparison of numerical and exact Lanczos vector spaces.

Main Results:

  • The sequence of numerical Lanczos vectors deviates from the true vector space spanned by exact Lanczos vectors.
  • Loss of orthogonality and reorthogonalization attempts do not fully resolve the numerical issues.
  • This deviation poses a significant challenge for theories like operator growth in quantum mechanics.

Conclusions:

  • The numerical inaccuracies in the Lanczos procedure for Krylov basis generation are not adequately addressed by standard reorthogonalization.
  • The escape of numerical vectors from the exact space has critical implications for understanding concepts like Krylov complexity and operator growth.
  • Further research is needed to develop robust numerical methods for Lanczos-based Krylov subspace methods.